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Building Archimedean Space
by Bogdan Stoica
Submission summary
Authors (as registered SciPost users):  Bogdan Stoica 
Submission information  

Preprint Link:  https://arxiv.org/abs/1809.01165v2 (pdf) 
Date submitted:  20210930 17:05 
Submitted by:  Stoica, Bogdan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
I propose that physical theories defined over finite places (including $p$adic fields) can be used to construct conventional theories over the reals, or conversely, that certain theories over the reals "decompose" over the finite places, and that this decomposition applies to quantum mechanics, field theory, gravity, and string theory, in both Euclidean and Lorentzian signatures. I present two examples of the decomposition: quantum mechanics of a free particle, and Euclidean twodimensional Einstein gravity. For the free particle, the finite place theory is the usual free particle $p$adic quantum mechanics, with the Hamiltonian obtained from the real one by replacing the usual derivatives with Vladimirov derivatives, and numerical coefficients with $p$adic norms. For Euclidean twodimensional gravity, the finite place objects mimicking the role of the spacetime are $\mathrm{SL}(\mathbb{Q}_p)$ BruhatTits trees. I furthermore propose quadratic extension BruhatTits trees as the finite place objects into which Lorentzian $\mathrm{AdS}_2$ decomposes, and BruhatTits buildings as a natural generalization to higher dimensions, with the same symmetry group on the finite and real sides for the manifolds and buildings corresponding to the vacuum state.
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Reports on this Submission
Report #1 by Lin Chen (Referee 1) on 202217 (Invited Report)
Report
$\textbf{Summary of the manuscript:}$
In this manuscript, the author proposes that the theories defined over the continuum spaces can be recovered from those defined over finite places such as $p$adic fields. And the author presents two examples to support this proposal: quantum mechanics of a free particle, and Euclidean two dimensional Einstein gravity.
In the $p$adic quantum mechanics part, the author introduces a parameter $\tau$, and considers a quadratic extension $Q_p[\sqrt{\tau}]$. Then sign functions $sgn_\tau$ are introduced to define multiplicative characters of $Q_p^{\times}$. And the Vladimirov derivative is defined by these multiplicative characters. After these preparation, the author writes down the $p$adic Schr$\ddot{o}$dinger equation and focuses on the free particle case. And the author finds that the position (momentum) space Archimedean propagators can be recovered by the product of the position (momentum) space $p$adic propagators, and the Archimedean plane waves are the multiplication of $p$adic plane waves.
In the Euclidean two dimensional Einstein gravity part, the author proposes that the BruhatTits tree is the $p$adic analogue of Euclidean $AdS_2$, and that a genus $g$ configuration of the Euclidean $AdS_2$ can be reconstructed from genus $g$ configurations of the BruhatTits tree. This proposal is mainly supported by the matching of partition functions of these configurations and the comparison of the geodesic lengths in $EAdS_2$ and in the BT tree.
In this manuscript, the author also tries to construct Lorentzian $AdS_2$ by the quadratic extension of BruhatTits tree. In this quadratic extension, edges are assigned spacelike and timelike labels. The author defines curvature and action on the tree, and works out the linearized Einstein equations on it.
$\textbf{Questions and comments:}$
$\textbf{Comment 1:}$
In the $p$adic quantum mechanics part, the connection between $p$adic theory and Archimedean theory is shown clearly. Though the results are not entirely new, the introduction of the parameter $\tau$ and sign functions $sgn_\tau$ is interesting, as can be seen from the condition $sgn_\tau(2m)=1$.
$\textbf{Question 2:}$
In the Euclidean $AdS_2$ part, the boundary points in $EAdS_2$ and in the BT tree share the same coordinate $x\in Q$. However the relation between bulk points in $EAdS_2$ and in the BT tree is still unclear. Is it possible to elaborate on this relation some more ?
$\textbf{Comment 3:}$
In the Lorentzian $AdS_2$ part, it is hard to say Lorentzian $AdS_2$ is reconstructed from this quadratic extension tree. The comparison between Lorentzian $AdS_2$ and the quadratic extension tree in section 5.6 is too rough. Important structure characterizing a Lorentzian spacetime, such as causal structure, is not clear even with the assignment of spacelike and timelike labels to each edge.
Though whether Lorentzian $AdS_2$ is reconstructed is doubtful, this attempt of constructing another type of tree may bring us some inspiration, and is worth being noticed.
$\textbf{Recommendation:}$
I recommend the manuscript to be published. Though the construction of Lorentzian $AdS_2$ is not so convincing, the structure of quadratic extension tree is new and may bring some inspiration. The idea of introducing a parameter $\tau$ and considering quadratic extension might turn out to be useful. It is desirable that the author elaborate some more on say causal structure of the Lorentzian tree, but this is not a necessary addition to render the paper publishable.